High omnidirectional reflector

ABSTRACT

A reflector, a method of producing same and a method of creating high omnidirectional reflection for a predetermined range of frequencies of incident electromagnetic energy for any angle of incidence and any polarization. The reflector includes a structure with a surface and a refractive index variation along the direction perpendicular to the surface while remaining nearly uniform along the surface. The structure is configured such that i) a range of frequencies exists defining a photonic band gap for electromagnetic energy incident along the perpendicular direction of said surface, ii) a range of frequencies exists defining a photonic band gap for electromagnetic energy incident along a direction approximately 90° from the perpendicular direction of said surface, and iii) a range of frequencies exists which is common to both of said photonic band gaps. In an exemplary embodiment, the reflector is configured as a photonic crystal.

[0001] This application claims priority from provisional applicationSer. No. 60/075,223 filed Feb. 19, 1998.

[0002] This invention was made with government support under 9400334-DRMawarded by the National Science Foundation. The government has certainrights in the invention.

BACKGROUND OF THE INVENTION

[0003] The invention relates to the field of photonic crystals, and inparticular to a dielectric high omnidirectional reflector.

[0004] Low-loss periodic dielectrics, or “photonic crystals”, allow thepropagation of electromagnetic energy, e.g., light, to be controlled inotherwise difficult or impossible ways. The existence of photonicbandgap in certain photonic crystals has given rise to the possibilitythat a photonic crystal can be a perfect mirror for light from anydirection, with any polarization, within a specified frequency range.Within the frequency range of photonic bandgaps, there are nopropagating solutions of Maxwell's equations inside a periodic medium.Consequently, a wave-front with a frequency within the gap which isincident upon the surface of such a crystal would be completelyreflected.

[0005] It is natural to assume that a necessary condition for suchomnidirectional reflection is that the photonic crystal exhibit acomplete three-dimensional photonic band-gap, i.e., a frequency rangewithin which there are no propagating solutions of Maxwell's equations.Such a photonic crystal would require periodic variations in dielectricconstant in all three dimensions. These crystals, if designed forinfrared or optical light, are difficult to fabricate, since the spatialperiods must be comparable to the wavelength of operation. This is thereason why, despite heroic experiments involving advanced lithographicmethods or self-assembling microstructures, most of the proposals forutilizing photonic crystals are in early stages of development.

SUMMARY OF THE INVENTION

[0006] It is therefore an object of the invention to provide adielectric structure that acts as a perfect mirror by exhibiting highomnidirectional reflection of energy regardless of polarization andincident angle.

[0007] It is a further object of the invention to provide aone-dimensionally periodic photonic crystal structure, such asmulti-layer films, that can exhibit complete reflection of radiation ina given frequency range for all incident angles and polarizations.

[0008] Accordingly, the invention provides a reflector, a method ofproducing same and a method of creating high omnidirectional reflectionfor a predetermined range of frequencies of incident electromagneticenergy for any angle of incidence and any polarization. The reflectorincludes a structure with a surface and a refractive index variationalong the direction perpendicular to the surface while remaining nearlyuniform along the surface. The structure is configured such that i) arange of frequencies exists defining a photonic band gap forelectromagnetic energy incident along the perpendicular direction ofsaid surface, ii) a range of frequencies exists defining a photonic bandgap for electromagnetic energy incident along a direction approximately90° from the perpendicular direction of said surface, and iii) a rangeof frequencies exists which is common to both of said photonic bandgaps. In one exemplary embodiment the reflector is configured as aphotonic crystal.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009]FIG. 1 is a schematic block diagram of an exemplary embodiment ofa high omnidirectional reflector in accordance with the invention;

[0010]FIG. 2 is a graph of the first three bands of an exemplarymultilayer film quarter-wave stack;

[0011]FIG. 3 is a graph showing the projected band structure for aquarter-wave stack with n₁=1, n₂=2;

[0012]FIG. 4 is a graph showing the projected band structure for aquarter-wave stack with the same ratio n₂/n₁=2 and n₁=1.7, n₂=3.4(α=1.7), and d₁=0.67a, d₂=0.33a, where a is the period;

[0013]FIG. 5 is a graph of the calculated spectra for a quarter-wavestack of ten films (n₁=1.7, n₂=3.4) for three angles of incidence; and

[0014]FIG. 6 is a contour plot of the range-midrange ratio for thefrequency range of high omnidirectional reflection, as n₁ and n₂/n₁ arevaried, for the maximizing value of d₁/a.

DETAILED DESCRIPTION OF THE INVENTION

[0015]FIG. 1 is a schematic block diagram of an exemplary embodiment ofa high omnidirectional reflector 100 in accordance with the invention.The reflector 100 is a one-dimensionally periodic photonic crystalhaving an index of refraction that is periodic in the y-coordinate,perpendicular to a surface 101, and consists of a repeating stack ofdielectric slabs 102, 104, which alternate in thickness from d₁ to d₂(in the illustrated embodiment d₁=d and d₂=1-d) and an index ofrefraction from n₁ to n₂. In the illustrated embodiment, d₁ and d₂ areassumed to be in the unit of period a. Only a few periods of such aperiodic system are illustrated. For a quarter-wave stack, n₁d₁=n₂d₂.The stacks are fabricated in a conventional manner on a substrate 106,e.g., silicon.

[0016]FIG. 1 also shows two orthogonal polarizations of incident light.An s-polarized wave has an electric field E perpendicular to the planeof incidence and a magnetic field B parallel to the plane of incidence.A p-polarized wave has an electric field parallel to the plane ofincidence and a magnetic field perpendicular to the plane of incidence.Since the medium is periodic in the y-direction (discrete translationalsymmetry) and homogeneous in the x- and z-directions (continuoustranslational symmetry), the electromagnetic modes can be characterizedin Bloch form by a wave vector k. In particular, ky is restricted to thefirst Brillouin zone −π/a<k_(y)<π/a, and k_(x) and k_(z) areunrestricted. One can suppose that k_(z)=0, k_(x)≧0 and n₂>n₁ withoutloss of generality. The allowed mode frequencies ω_(n) for each choiceof k constitute the band structure of the crystal. The continuousfunctions ω_(n)(k), for each n, are the photonic bands.

[0017]FIG. 2 is a graph of the first three bands of an exemplarymultilayer film quarter-wave stack with n₁=1, n₂=2, as a function ofk_(y), for the special case k_(x)=0 (normal incidence). The thicknesseswere chosen to be d₁=0.67 and d₂=0.33. For k_(x)=0, there is nodistinction between s- and p-polarized waves. There is a wide frequencygap between the first and second bands. This splitting arises from thedestructive interference of the waves which are transmitted andreflected at each interface. It will be appreciated that the frequencyhas been expressed in units of c/a, where c is the speed of light in theambient medium and a=d₁+d₂.

[0018] Any one-dimensional photonic crystal, as defined by a varyingindex function n(y) that in the illustrated case is periodic will have anon-zero gap for k_(x)=0. Within it there are no propagating modes, so awave with its frequency falling in the range of the gap, if incidentnormal to the surface of such a crystal, will be reflected.

[0019] For k_(x)>0 (an arbitrary direction of propagation) it isconvenient to examine the projected band structure, which is shown inFIG. 3 for the same medium as in FIG. 2, a quarter-wave stack with n₁=1,n₂=2. To make this plot, first the bands ω_(n)(k_(x), k_(y)) for thestructure were computed, using a numerical method for solving Maxwell'sequations in a periodic medium. For each value of k_(x), the modefrequencies ω_(n), for all possible values of k_(y) were plotted. Thus,in the gray regions there are electromagnetic modes for some values ofk_(y), whereas in the white regions there are no electromagnetic modes,regardless of ky. The s-polarized modes are plotted to the right of theorigin, and the p-polarized modes to the left. Frequencies are reportedin units of c/a.

[0020] The shape of the projected band structure for a multilayer filmcan be understood intuitively. At k_(x)=0, the normal-incidence bandgapof FIG. 2 is recovered. This range of frequencies is enclosed by dashedlines. As k_(x)>0, the bands curve upwards in frequency, as thecondition for destructive interference shifts to shorter wavelengths. Ask_(x)→∞, the frequency width of the gray regions shrinks until theybecome lines. In this regime the modes are largely confined to the slabswith the higher index of refraction. For large k_(x) they are very wellconfined and do not couple between layers (independent of k_(y)). Theyare approximately planar waveguide modes, so the dispersion relationapproaches ω=ck_(x)/n₂ asymptotically.

[0021] One obvious feature of FIG. 3 is that there is no completebandgap. For any frequency, there exists a wave-vector and an associatedelectromagnetic mode corresponding to that frequency. Thenormal-incidence bandgap 300 (enclosed by the dashed lines) is crossedby modes with k_(x)>0. This is a general feature of one-dimensionalphotonic crystals.

[0022] However, the absence of a complete band-gap does not precludeomnidirectional reflection. The criterion is not that there be nopropagating states within the crystal; rather, the criterion is thatthere be no propagating states that may couple to an incidentpropagating wave. This is equivalent to the existence of a frequencyrange in which the projected band structures of the crystal and theambient medium have no overlap.

[0023] The two diagonal black lines 302, 304 in FIG. 3 are the “lightlines” ω=ck_(x). The electromagnetic modes in the ambient medium (air)obey ω=c(k_(x) ²+k_(y) ²)^(½), where c is the speed of light in theambient medium, so generally ω>ck_(x). The whole region above the soliddiagonal “light-lines” ω>ck_(x) is filled with the projected bands ofthe ambient medium.

[0024] For a semi-infinite crystal occupying y<0 and an ambient mediumoccupying y>0, the system is no longer periodic in the y-direction (notranslational symmetry) and the electromagnetic modes of the system canno longer be classified by a single value of k_(y). They must be writtenas a weighted sum of plane waves with all possible k_(y). However, k_(x)is still a valid symmetry label. The angle of incidence θ upon theinterface at y=0 is related to k_(x) by ωsinθ=ck_(x).

[0025] For there to be any transmission through the semi-infinitecrystal at a particular frequency, there must be an electromagnetic modeavailable at that frequency which is extended for both y>0 and y<0. Sucha mode must be present in the projected photonic band structures of boththe crystal and the ambient medium. The only states that could bepresent in the semi-infinite system that were not present in the bulksystem are surface states, which decay exponentially in both directionsaway from the surface, and are therefore irrelevant to the transmissionof an external wave. Therefore, the criterion for high omnidirectionalreflection is that there are no states in common between the projectedbands of the ambient medium and those of the crystal, i.e., there existsa frequency zone in which the projected bands of the crystal have nostates with ω>ck_(x).

[0026] It can be seen from FIG. 3 that there is such a frequency zone(0.36c/a to 0.45c/a) for s-polarized waves. The zone is bounded above bythe normal-incidence bandgap, and below by the intersection of the topof the first gray region with the light line. The top edge of the firstgray region is the dispersion relation for states with k_(y)=π/a.

[0027] The lowest two p-bands cross at a point above the line ω=ck_(x),preventing the existence of such a frequency zone. This crossing occursat the Brewster angle θ_(B)=tan⁻¹(n₂/n₁), at which there is noreflection of p-polarized waves at any interface. At this angle there isno coupling between waves with k_(y) and −k_(y), a fact which permitsthe band-crossing to occur. As a result, the bands curve upwards morerapidly.

[0028] This difficulty vanishes when the bands of the crystal arelowered relative to those of the ambient medium, by raising the indicesof refraction of the dielectric films. For example, by multiplying theindex of refraction n(y) by a constant factor α>1, all of thefrequencies of the electromagnetic modes are lowered by the same factorα.

[0029]FIG. 4 is a graph showing the projected band structure for anexemplary quarter-wave stack with the same ratio n₂/n₁=2 and n₁=1.7,n₂=3.4 (α=1.7), and d₁=0.67, d₂=0.33. In this case there is a frequencyzone in which the projected bands of the crystal and ambient medium donot overlap, namely from the point 400 (ωa/2πc=0.21) to the point 402(ωa/2πc=0.27). This zone is bounded above by the normal-incidencebandgap and below by the intersection of the top of the first grayregion for p-polarized waves with the light-line 404. While theillustrated embodiments of the invention will be described utilizing asilicon-silicon dioxide materials system, the invention can befabricated with other materials systems.

[0030] Between the frequencies corresponding to the points 400 and 402,there will be total reflection from any incident angle for eitherpolarization. For a finite number of films, the transmitted light willdiminish exponentially with the number of films. The calculatedtransmission spectra, for a finite system of ten films (five periods),are plotted in FIG. 5 for various angles of incidence, e.g., from 0° toapproximately 90°. The calculations were performed using transfermatrices. The stop band shifts to higher frequencies with more obliqueangles, but there is a region of overlap which remains intact for allangles.

[0031]FIG. 5 is a graph of the calculated spectra for a quarter-wavestack of ten films (n1=1.7, n2=3.4) for three angles of incidence. Thesolid curves correspond to p-polarized waves, and the dashed curvescorrespond to s-polarized waves. The overlapping region of highreflectance corresponds to the region between the points 400 and 402 ofFIG. 4. While the illustrated embodiment describes the characteristicsof a structure having a ten-layer film of silicon and silicon dioxide,it will be appreciated that a reflector of the invention can be achievedwith other multilayer arrangements or other material systems withappropriate index contrasts.

[0032] The criterion for high omnidirectional reflection (thenon-overlap of the projected bands of both crystal and ambient medium)applies for a general function n(y) that is not necessarily periodic.For the special case of a multilayer film it is possible to derive anexplicit form of the band structure function ω_(n)(k_(x),k_(y)) and useit to investigate systematically the frequency zone of directionalreflection, if any, which results from a given choice of n₁, n₂, d₁ andd₂.

[0033] The graphical criterion for high omnidirectional reflection, asshown in FIG. 4, is that the point 400 (the intersection of the lightline 404 and the first p-polarized band at k_(y)=π/a) be lower than thepoint 402 (the second band at k_(x)=0, k_(y)=π/a). Symbolically,$\begin{matrix}{{\omega_{p1}\left( {{k_{x} = \frac{\omega_{p1}}{c}},{k_{y} = \frac{\pi}{a}}} \right)} < {\omega_{p2}\left( {{k_{x} = 0},{k_{y} = \frac{\pi}{a}}} \right)}} & (1)\end{matrix}$

[0034] where ω_(pn)(k_(x), k_(y)) is the p-polarized band structurefunction for the multilayer film. It will be appreciated that the leftside is a self-consistent solution for the frequency ω_(p1). Thedifference between these two frequencies is the range of highomnidirectional reflection.

[0035] For a multilayer film, the dispersion relation ω_(n)(k_(x),k_(y))may be derived by computing the eigenvalues Λ of the transfer matrixassociated with one period of the film at a particular frequency andincident angle. When Λ=exp(ik_(y)a) with k_(y) real, there is apropagating mode at that frequency and angle. The dispersion relationω_(n)(k_(x),k_(y)) is governed by the transcendental equation:$\begin{matrix}{{{\left( {1 + \frac{A}{2}} \right){\cos \left\lbrack {\left( {\beta_{2} + \beta_{1}} \right)\omega} \right\rbrack}} - {\frac{A}{2}{\cos \left\lbrack {\left( {\beta_{2} - \beta_{1}} \right)\omega} \right\rbrack}}} = {\cos \left( {k_{y}a} \right)}} & (2)\end{matrix}$

[0036] Here β_(1,2)=(d_(1,2)/c){square root}n_(1,2) ²−sin²θ is definedfor each film. The polarization-dependent constant A is defined by:$\begin{matrix}{A = \frac{\left( {r_{1} - r_{2}} \right)^{2}}{2r_{1}r_{2}}} & (3) \\{r_{1,2} = \begin{Bmatrix}\sqrt{n_{1,2}^{2} - {\sin^{2}\theta}} & \left( {s - {polarized}} \right) \\\frac{n_{1,2}}{\sqrt{n_{1,2}^{2} - {\sin^{2}\theta}}} & \left( {p - {polarized}} \right)\end{Bmatrix}} & (4)\end{matrix}$

[0037] These results may be used to evaluate the criterion as expressedin equation (1). The roots of equation (2) may be found numerically, fora given k_(y) and θ=asin(ck_(x)/ω). The frequency range (if any) ofomidirectional reflection, according to equation (1), is between thefirst root of equation (2) for p-polarized waves with k_(y)=π/a andθ=π/2 (point 400 of FIG. 4), and the second root for k_(y)=π/a and θ=0(point 402).

[0038] The frequency range has been calculated (when it exists) for acomprehensive set of film parameters. Since all the mode wavelengthsscale linearly with d₁+d₂=a, only three parameters need to be consideredfor a multilayer film: n₁, n₂, and d₁/a. To quantify the range of highomnidirectional reflection [ω₁, ω₂] in a scale-independent manner, the“range-midrange ratio” is defined as (ω₂-ω₁)/[(1/2)(ω₁+ω₂)].

[0039] For each choice of n₁ and n₂/n₁, there is a value of dl/a thatmaximizes the range-midrange ratio. That choice may be computednumerically. FIG. 6 is a contour plot of the range-midrange ratio forthe frequency range of high omnidirectional reflection, as n₁ and n₂/n₁are varied, for the maximizing value of d₁/a (solid contours). Thedashed curve is the 0% contour for the case of a quarter-wave stack. Forthe general case of an ambient medium with index n₀≠1, the abscissabecomes n₁/n₀. This plot shows the largest possible range-midrange ratioachievable with n₁ and n₂ fixed.

[0040] An approximate analytic expression for the optimal zone of highomnidirectional reflection may be derived: $\begin{matrix}{{\frac{\Delta \quad \omega}{2c} = {\frac{a\quad {\cos \left( {- \sqrt{\frac{A - 2}{A + 2}}} \right)}}{{d_{1}n_{1}} + {d_{2}n_{2}}} - \frac{a\quad {\cos \left( {- \sqrt{\frac{B - 2}{B + 2}}} \right)}}{{d_{1}\sqrt{n_{1}^{2} - 1}} + {d_{2}\sqrt{n_{2}^{2} - 1}}}}}{where}} & (5) \\{{A \equiv {\frac{n_{2}}{n_{1}} + \frac{n_{1}}{n_{2}}}},\quad {B \equiv {\frac{n_{2}\sqrt{n_{1}^{2} - 1}}{n_{1}\sqrt{n_{2}^{2} - 1}} + \frac{n_{1}\sqrt{n_{2}^{2} - 1}}{n_{2}\sqrt{n_{1}^{2} - 1}}}}} & (6)\end{matrix}$

[0041] Numerically this is found to be an excellent approximation forthe entire range of parameters depicted in FIG. 6 including the case ofa quarter-wave stack.

[0042] It can be seen from FIG. 6 that, for high omnidirectionalreflection, the index ratio should be reasonably high (n₁/n₂>1.5) andthe indices themselves be somewhat higher (n₁/n₀≧1.5) than that of theambient medium. The former condition increases the band splittings, andthe latter depresses the frequency of the Brewster crossing. An increasein either factor can partially compensate for the other. The materialsshould also have a long absorption length for the frequency range ofinterest, especially at grazing angles, where the path length of thereflected light along the crystal surface is long.

[0043] For example, for light with a wavelength of 1.5 μm, silicondioxide has n₁=1.44 and silicon has n₂=3.48=2.42n₁. From FIG. 6, thiscorresponds to a range-midrange ratio of about 27%. Likewise, forGaAs/Al₂O₃ multilayers (n₁=1.75, n₂=3.37=1.93n₁), the range-midrangeratio is about 24%.

[0044] In practice, the optimization of d₁/a results in a gap size veryclose to the gap size that would result from a quarter-wave stacked withthe same indices d₁/a=n₂/(n₂+n₁). The 0% contour for quarter-wave stacksis plotted in FIG. 6 as a dashed line, which is very close to theoptimized 0% contour.

[0045] With this in mind, an approximation to equation (2) may bederived for films which are nearly quarter-wave stacks. In that limitβ₂-β₁≈0, so the second cosine in equation (2) is approximately 1. Inthis approximation the frequency of the band edge at ky=π/a is:$\begin{matrix}{\omega \approx {\frac{1}{\beta_{1} + \beta_{2}}a\quad {\cos \left\lbrack \frac{\frac{A}{2} + 1}{\frac{A}{2} - 1} \right\rbrack}}} & (7)\end{matrix}$

[0046] using the same notion as in equations (3) and (4). This frequencycan be computed for the cases θ=0 and θ=π/2. If the difference betweenthese two frequencies is positive, there will be omnidirectionalreflection for any frequency between them.

[0047] The invention demonstrates that, even though it is not possiblefor a one-dimensional photonic crystal to have a complete bandgap, it isstill possible to achieve reflection of ambient light regardless ofincident angle or polarization. This happens whenever the projectedbands of the crystal and ambient medium have no overlap within somerange of frequencies.

[0048] This constraint is not unrealistic, even for the most common sortof one-dimensional photonic crystal, the multilayer film. As can be seenin FIG. 6, what is required is that the index ratio be reasonably high(n₂/n₁>1.5) and the indices themselves be somewhat higher than that ofthe ambient medium (n₁/n₀>1.5). An increase in either factor canpartially compensate for the other. They should also have a relativelylong absorption length for the frequency range of interest. Suchmaterials, and the technology required to deposit them in multiplelayers, are conventional. To achieve high omnidirectional reflection,therefore, it is not necessary to use more elaborate systems such asmultiple interleaving stacks, materials with special dispersionproperties, or fully three-dimensional photonic crystals.

[0049] The optical response of a particular dielectric multilayer filmcan be predicted using the characteristic matrix method. In this method,a 2×2 unitary matrix is constructed for each layer. This matrixrepresents a mapping of the field components from one side of the layerto the other. To successfully predict the optical response of amultilayer film the characteristic matrix for each layer needs to becalculated. The form of the characteristic matrix for the j^(th) layeris $\begin{matrix}{{{m^{g}(\theta)}_{j} = {\begin{bmatrix}{\cos \quad \beta_{j}} & {{- \frac{i}{p_{j}^{g}}}\sin \quad \beta_{j}} \\{{- {ip}_{j}^{g}}\sin \quad \beta_{j}} & {\cos \quad \beta_{j}}\end{bmatrix}\quad \left( {{g = {TE}},{TM}} \right)}}{\beta_{j} = {{kh}_{j}\sqrt{n_{j}^{2} - {{snell}(\theta)}^{2}}}}{{{snell}(\theta)} = {n_{0}\sin \quad \theta_{0}}}{p_{j}^{g} = \left\{ \begin{matrix}\sqrt{n_{j}^{2} - {{snell}\quad (\theta)^{2}}} & {g = {TE}} \\\frac{\sqrt{n_{j}^{2} - {{snell}\quad (\theta)^{2}}}}{n_{j}^{2}} & {g = {TM}}\end{matrix} \right.}} & (8)\end{matrix}$

[0050] where n_(j) is the index of refraction, and h_(j) is thethickness of the J^(th) layer, θ₀ is the angle between the incident waveand the normal to the surface and n₀ is the index of the initial medium,e.g., air.

[0051] The matrices are then multiplied to give the film'scharacteristic matrix $\begin{matrix}{{M^{g}(\theta)} = {\prod\limits_{j = 1}^{N}\quad {m_{j}^{g}\left( {g = {{TM}\quad {or}\quad {TE}}} \right)}}} & (9)\end{matrix}$

[0052] which in turn can be used to calculate the reflectivity for agiven polarization and angle of incidence, $\begin{matrix}{{R^{g}(\theta)} = {\frac{{\left( {{M_{11}^{g}(\theta)} + {{M_{12}^{g}(\theta)}p_{1}^{g}}} \right)p_{0}^{g}} - \left( {{M_{21}^{g}(\theta)} + {{M_{22}^{g}(\theta)}p_{1}^{g}}} \right)}{{\left( {{M_{11}^{g}(\theta)} + {{M_{12}^{g}(\theta)}p_{1}^{g}}} \right)p_{0}^{g}} + \left( {{M_{21}^{g}(\theta)} + {{M_{22}^{g}(\theta)}p_{1}^{g}}} \right)}}^{2}} & (10)\end{matrix}$

[0053] where p^(g) ₀ contains information about the index of the mediumand angle of incidence on one side of the multilayer film and p^(g) ₁contains information about the index of the medium and angle ofincidence on the other.

[0054] To construct a reflector exhibiting a reflectivity R of a minimalprescribed value for all angles of incidence and both polarizations oneneeds to (1) satisfy the criteria for omnidirectional reflection, and(2) solve equation (10) for θ=89.9°, g=TM and R™ (89.9)=R.

[0055] Although the invention has been illustrated by using multilayeredfilms, the invention as described can apply generally to any periodicdielectric function n(y), or even an aperiodic dielectric function n(y).What is required is that n(y) leads to photonic bandgaps along variousdirections such that there exists a zone of frequencies in which theprojected bands of the dielectric structure and ambient media do notoverlap. Such a requirement can also be satisfied by a photonic crystalwith two- or three-dimensionally periodic index contrasts, which haveincomplete bandgaps.

[0056] However, the absence of a complete bandgap does have physicalconsequences. In the frequency range of high omnidirectional reflection,there exist propagating solutions of Maxwell's equations, but they arestates with ω<ck_(x), and decrease exponentially away from the crystalboundary. If such a state were launched from within the crystal, itwould propagate to the boundary and reflect, just as in total internalreflection.

[0057] Likewise, although it might be arranged that the propagatingstates of the ambient medium do not couple to the propagating states ofthe crystal, any evanescent states in the ambient medium will couple tothem. For this reason, a point source of waves placed very close (d<λ)to the crystal surface could indeed couple to the propagating state ofthe crystal. Such restrictions, however, apply only to a point sourceand can be easily overcome by simply adding a low index cladding layerto separate the point source from the film surface.

[0058] Many potential applications are envisioned for such a highomnidirectional reflector or mirror. For example, in the infrared,visible, or ultraviolet regimes, high omnidirectional reflectors couldserve as a frequency-selective mirrors for laser beams orhighly-reflective coatings on focusing instruments. These would beeffective for light that is incident from any angle, instead of just afinite range around a fixed design angle.

[0059] The invention can also be utilized in coatings with infraredmirrors to keep heat in or out of the items coated, e.g., walls,windows, clothes, etc. The mirrors can be cut into small flakes andmixed with paint or fabrics to allow for application to the desireditems.

[0060] The reflector of the invention could be used in improvingthermo-photovoltaic cells that trap waste heat and convert it intoenergy. The reflector of the invention can also be made to reflect radiowaves and thus can be used to boost performance of radio devices such ascellular telephones.

[0061] Although the present invention has been shown and described withrespect to several preferred embodiments thereof, various changes,omissions and additions to the form and detail thereof, may be madetherein, without departing from the spirit and scope of the invention.

What is claimed is:
 1. A method of producing a reflector which exhibitshigh omnidirectional reflection for a predetermined range of frequenciesof incident electromagnetic energy for any angle of incidence and anypolarization, comprising: configuring a structure with a surface and arefractive index variation along the direction perpendicular to saidsurface while remaining nearly uniform along the surface, said structureconfigured such that i) a range of frequencies exists defining aphotonic band gap for electromagnetic energy incident along theperpendicular direction of said surface, ii) a range of frequenciesexists defining a photonic band gap for electromagnetic energy incidentalong a direction approximately 90° from the perpendicular direction ofsaid surface, and iii) a range of frequencies exists which is common toboth of said photonic band gaps.
 2. The method of claim 1, wherein stepiii) comprises a range of maximum frequencies that exists in common toboth of said photonic band gaps.
 3. The method of claim 1, whereinranges of frequencies exist defining photonic band gaps forelectromagnetic energy incident along directions between 0° andapproximately 90° from the perpendicular direction of said surface. 4.The method of claim 1, wherein said structure is configured as aphotonic crystal.
 5. The method of claim 4, wherein said structure isconfigured as a one dimensionally periodic dielectric structure.
 6. Themethod of claim 4, wherein said periodic dielectric structure comprisesperiodic units each having two or more layers.
 7. The method of claim 6,wherein said periodic units comprise layers of silicon and silicondioxide.
 8. The method of claim 6, wherein said periodic units compriselayers of GaAs and Al_(x)O_(y).
 9. The method of claim 6, wherein thezone for high omnidirectional reflection is$\frac{\Delta \quad \omega}{2c} = {\frac{a\quad {\cos \left( {- \sqrt{\frac{A - 2}{A + 2}}} \right)}}{{d_{1}n_{1}} + {d_{2}n_{2}}} - \frac{a\quad {\cos \left( {- \sqrt{\frac{B - 2}{B + 2}}} \right)}}{{d_{1}\sqrt{n_{1}^{2} - 1}} + {d_{2}\sqrt{n_{2}^{2} - 1}}}}$where${A \equiv {\frac{n_{2}}{n_{1}} + \frac{n_{1}}{n_{2}}}},\quad {B \equiv {\frac{n_{2}\sqrt{n_{1}^{2} - 1}}{n_{1}\sqrt{n_{2}^{2} - 1}} + {\frac{n_{1}\sqrt{n_{2}^{2} - 1}}{n_{2}\sqrt{n_{1}^{2} - 1}}.}}}$


10. The method of claim 6, wherein the layer thickness of materials offirst and second layers with respective indices of refraction definedwith respect to the ambient are chosen such that Δω is greater thanzero.
 11. The method of claim 1, wherein said structure is configuredwith a continuous variation in refractive index.
 12. The method of claim1, wherein said structure is configured as an aperiodic dielectricstructure.
 13. The method of claim 1, wherein said reflector exhibitsgreater than 99% reflectivity.
 14. A high omnidirectional reflectorwhich exhibits reflection for a predetermined range of frequencies ofincident electromagnetic energy for any angle of incidence and anypolarization, comprising: a structure with a surface and a refractiveindex variation along the direction perpendicular to said surface whileremaining nearly uniform along the surface, said structure configuredsuch that i) a range of frequencies exists defining a photonic band gapfor electromagnetic energy incident along the perpendicular direction ofsaid surface, ii) a range of frequencies exists defining a photonic bandgap for electromagnetic energy incident along a direction approximately90° from the perpendicular direction of said surface, and iii) a rangeof frequencies exists which is common to both of said photonic bandgaps.
 15. The method of claim 14, wherein item iii) comprises a range ofmaximum frequencies that exists in common to both of said photonic bandgaps.
 16. The reflector of claim 14, wherein ranges of frequencies existdefining photonic band gaps for electromagnetic energy incident alongdirections between 0° and approximately 90° from the perpendiculardirection of said surface.
 17. The reflector of claim 14, wherein saidstructure is configured as a photonic crystal.
 18. The reflector ofclaim 17, wherein said structure is configured as a one dimensionallyperiodic dielectric structure.
 19. The reflector of claim 17, whereinsaid periodic dielectric structure comprises periodic units each havingtwo or more layers.
 20. The reflector of claim 19, wherein said periodicunits comprise layers of silicon and silicon dioxide.
 21. The reflectorof claim 19, wherein said periodic units comprise layers of GaAs andAl_(x)O_(y).
 22. The reflector of claim 19, wherein the zone for highomnidirectional reflection is$\frac{\Delta \quad \omega}{2c} = {\frac{a\quad {\cos \left( {- \sqrt{\frac{A - 2}{A + 2}}} \right)}}{{d_{1}n_{1}} + {d_{2}n_{2}}} - \frac{a\quad {\cos \left( {- \sqrt{\frac{B - 2}{B + 2}}} \right)}}{{d_{1}\sqrt{n_{1}^{2} - 1}} + {d_{2}\sqrt{n_{2}^{2} - 1}}}}$where${A \equiv {\frac{n_{2}}{n_{1}} + \frac{n_{1}}{n_{2}}}},\quad {B \equiv {\frac{n_{2}\sqrt{n_{1}^{2} - 1}}{n_{1}\sqrt{n_{2}^{2} - 1}} + {\frac{n_{1}\sqrt{n_{2}^{2} - 1}}{n_{2}\sqrt{n_{1}^{2} - 1}}.}}}$


23. The reflector of claim 19, wherein the layer thickness of materialsof first and second layers with respective indices of refraction definedwith respect to the ambient are chosen such that Δω is greater thanzero.
 24. The method of claim 14, wherein said structure is configuredwith a continuous variation in refractive index.
 25. The method of claim14, wherein said structure is configured as an aperiodic dielectricstructure.
 26. The method of claim 14, wherein said reflector exhibitsgreater than 99% reflectivity.
 27. A method of creating highomnidirectional reflection for a predetermined range of frequencies ofincident electromagnetic energy for any angle of incidence and anypolarization, comprising: providing a structure with a surface and arefractive index variation along the direction perpendicular to saidsurface while remaining nearly uniform along the surface, said structureconfigured such that i) a range of frequencies exists defining aphotonic band gap for electromagnetic energy incident along theperpendicular direction of said surface, ii) a range of frequenciesexists defining a photonic band gap for electromagnetic energy incidentalong a direction approximately 90° from the perpendicular direction ofsaid surface, and iii) a range of frequencies exists which is common toboth of said photonic band gaps.
 28. The method of claim 27, whereinitem iii) comprises a range of maximum frequencies that exists in commonto both of said photonic band gaps.
 29. The method of claim 27, whereinranges of frequencies exist defining photonic band gaps forelectromagnetic energy incident along directions between 0° andapproximately 90° from the perpendicular direction of said surface. 30.The method of claim 27, wherein said structure is configured as aphotonic crystal.
 31. The method of claim 30, wherein said structure isconfigured as a one dimensionally periodic dielectric structure.
 32. Themethod of claim 30, wherein said periodic dielectric structure comprisesperiodic units each having two or more layers.
 33. The method of claim32, wherein said periodic units comprise layers of silicon and silicondioxide.
 34. The method of claim 32, wherein said periodic unitscomprise layers of GaAs and Al_(x)O_(y).
 35. The method of claim 32,wherein the zone for high omnidirectional reflection is$\frac{\Delta \quad \omega}{2c} = {\frac{a\quad {\cos\left( {- \sqrt{\frac{A - 2}{A + 2}}} \right.}}{{d_{1}n_{1}} + {d_{2}n_{2}}} - \frac{a\quad {\cos \left( {- \sqrt{\frac{B - 2}{B + 2}}} \right)}}{{d_{1}\sqrt{n_{1}^{2} - 1}} + {d_{2}\sqrt{n_{2}^{2} - 1}}}}$where${A \equiv {\frac{n_{2}}{n_{1}} + \frac{n_{1}}{n_{2}}}},{B \equiv {\frac{n_{2}\sqrt{n_{1}^{2} - 1}}{n_{1}\sqrt{n_{2}^{2} - 1}} + {\frac{n_{1}\sqrt{n_{2}^{2} - 1}}{n_{2}\sqrt{n_{1}^{2} - 1}}.}}}$


36. The method of claim 32, wherein the layer thickness of materials offirst and second layers with respective indices of refraction definedwith respect to the ambient are chosen such that Δω is greater thanzero.
 37. The method of claim 27, wherein said structure is configuredwith a continuous variation in refractive index.
 38. The method of claim27, wherein said structure is configured as an aperiodic dielectricstructure.
 39. The method of claim 27, wherein the omnidirectionalachieved is greater than 99%.
 40. A method for producing an alldielectric omnidirectional reflector which exhibits omnidirectionalreflection that is greater than 95% for a predetermined range offrequencies of incident electromagnetic energy of any angle of incidenceand any polarization comprising: providing a structure with a surfaceand a refractive index variation along the direction perpendicular tothe said surface while remaining nearly uniform along the surface saidsurface configured such that (i) a range of frequencies exists defininga reflectivity range which is higher than 99% for EM energy incidentalong the perpendicular direction of the said surface, (ii) a range offrequencies exists defining a reflectivity range which is higher than99% for EM energy incident a direction approximately 90° from theperpendicular direction of the said surface, and (iii) a range offrequencies exists which is common to both of said reflectivity ranges.41. The method of claim 40, wherein the reflectivity is greater than 96%42. The method of claim 40, wherein the reflectivity is greater than 97%43. The method of claim 40, wherein the reflectivity is greater than 98%44. The method of claim 40, wherein the reflectivity is greater than 99%